We consider a sequence of single-server queueing models operating under a
service policy that incorporates batches into processor sharing: arriving jobs
build up behind a gate while waiting to begin service, while jobs in front of
the gate are served according to processor sharing. When they have been
completed, the waiting jobs move in front of the gate and the cycle repeats. We
model this system with a pair of measure valued processes describing the jobs
in front of and behind the gate. Under mild asymptotically critical conditions
and a law-of-large-numbers scaling, we prove that the pair of measure-valued
processes converges in distribution to an easily described limit, which has an
interesting periodic dynamics